MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mon1pval Structured version   Unicode version

Theorem mon1pval 20069
Description: Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pval.p  |-  P  =  (Poly1 `  R )
uc1pval.b  |-  B  =  ( Base `  P
)
uc1pval.z  |-  .0.  =  ( 0g `  P )
uc1pval.d  |-  D  =  ( deg1  `  R )
mon1pval.m  |-  M  =  (Monic1p `  R )
mon1pval.o  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mon1pval  |-  M  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  ) }
Distinct variable groups:    B, f    D, f    .1. , f    R, f    .0. , f
Allowed substitution hints:    P( f)    M( f)

Proof of Theorem mon1pval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 mon1pval.m . 2  |-  M  =  (Monic1p `  R )
2 fveq2 5731 . . . . . . . 8  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
3 uc1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
42, 3syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
54fveq2d 5735 . . . . . 6  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  ( Base `  P
) )
6 uc1pval.b . . . . . 6  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2488 . . . . 5  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  B )
84fveq2d 5735 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
9 uc1pval.z . . . . . . . 8  |-  .0.  =  ( 0g `  P )
108, 9syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1110neeq2d 2617 . . . . . 6  |-  ( r  =  R  ->  (
f  =/=  ( 0g
`  (Poly1 `  r ) )  <-> 
f  =/=  .0.  )
)
12 fveq2 5731 . . . . . . . . . 10  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
13 uc1pval.d . . . . . . . . . 10  |-  D  =  ( deg1  `  R )
1412, 13syl6eqr 2488 . . . . . . . . 9  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1514fveq1d 5733 . . . . . . . 8  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  f )  =  ( D `  f ) )
1615fveq2d 5735 . . . . . . 7  |-  ( r  =  R  ->  (
(coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( (coe1 `  f ) `  ( D `  f ) ) )
17 fveq2 5731 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
18 mon1pval.o . . . . . . . 8  |-  .1.  =  ( 1r `  R )
1917, 18syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2016, 19eqeq12d 2452 . . . . . 6  |-  ( r  =  R  ->  (
( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( 1r `  r
)  <->  ( (coe1 `  f
) `  ( D `  f ) )  =  .1.  ) )
2111, 20anbi12d 693 . . . . 5  |-  ( r  =  R  ->  (
( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) )  <->  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  )
) )
227, 21rabeqbidv 2953 . . . 4  |-  ( r  =  R  ->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) ) }  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
23 df-mon1 20058 . . . 4  |- Monic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) ) } )
24 fvex 5745 . . . . . 6  |-  ( Base `  P )  e.  _V
256, 24eqeltri 2508 . . . . 5  |-  B  e. 
_V
2625rabex 4357 . . . 4  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  e.  _V
2722, 23, 26fvmpt 5809 . . 3  |-  ( R  e.  _V  ->  (Monic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
28 fvprc 5725 . . . 4  |-  ( -.  R  e.  _V  ->  (Monic1p `  R )  =  (/) )
29 ssrab2 3430 . . . . . 6  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  B
30 fvprc 5725 . . . . . . . . . 10  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
313, 30syl5eq 2482 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  P  =  (/) )
3231fveq2d 5735 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  ( Base `  (/) ) )
336, 32syl5eq 2482 . . . . . . 7  |-  ( -.  R  e.  _V  ->  B  =  ( Base `  (/) ) )
34 base0 13511 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3533, 34syl6eqr 2488 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3629, 35syl5sseq 3398 . . . . 5  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  (/) )
37 ss0 3660 . . . . 5  |-  ( { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  (/)  ->  { f  e.  B  |  (
f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  =  (/) )
3836, 37syl 16 . . . 4  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  =  (/) )
3928, 38eqtr4d 2473 . . 3  |-  ( -.  R  e.  _V  ->  (Monic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
4027, 39pm2.61i 159 . 2  |-  (Monic1p `  R
)  =  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }
411, 40eqtri 2458 1  |-  M  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   ` cfv 5457   Basecbs 13474   0gc0g 13728   1rcur 15667  Poly1cpl1 16576  coe1cco1 16579   deg1 cdg1 19982  Monic1pcmn1 20053
This theorem is referenced by:  ismon1p  20070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-slot 13478  df-base 13479  df-mon1 20058
  Copyright terms: Public domain W3C validator